Question 27
A grid board with $a$ rows and $b$ columns ($a \ge 2, b \ge 2$) is given, together with dominoes formed by two $1 \times 1$ squares, which have the number $1$ written on one of the squares and the number $-1$ on the other. The board must be covered with dominoes, without gaps or overlaps and without sticking out of the board, in such a way that the product of the numbers written in each row is negative and the product of the numbers written in each column is positive. Determine the values of $a$ and $b$ for which this is possible. For these values, show how to place the dominoes. For the other values of $a$ and $b$, explain why it is not possible to place the dominoes.